Abstract

We mainly present several equivalent characterizations of the strong metric subregularity of the Mordukhovich subdifferential for an extended-real-valued lower semicontinuous, prox-regular, and subdifferentially continuous function acting on an Asplund space.

1. Introduction

This work is devoted to characterizations of the strong metric subregularity of the Mordukhovich subdifferential for an extended-real-valued prox-regular subdifferentially continuous function defined on an Asplund space.

Aragón and Geoffroy [1] have established remarkable equivalences between various metric regularity properties (including the (strong) metric subregularity and the (strong) metric regularity) of the subgradient mapping and a local quadratic growth condition for a proper convex lower semicontinuous function defined on a Hilbert space . Later, Drusvyatskiy and Lewis [2] proved that the characterization established in [1] for the strong metric regularity remains valid for the Mordukhovich subdifferential at for of a lower semicontinuous (not necessarily convex) function defined on when the function is subdifferentially continuous at for . Subsequently, Mordukhovich and Nghia [3] generalized this result to Asplund spaces without the assumption of subdifferential continuity [3, Corollary 3.3]

Recently, Aragón and Geoffroy [4] established the equivalence between the strong metric subregularity of the subgradient mapping for a proper lower semicontinuous (l.s.c.) convex function defined on a Banach space and the following quadratic growth condition:
where is a neighborhood of and . They raised the question whether the bound can be improved. Drusvyatskiy et al. [5] extended the above equivalence to the case, where the function is a proper lower semicontinuous function defined on an Asplund space and gave the affirmative answer to the above question via showing that the constant in (1) may be chosen arbitrarily in .

In the present paper, motivated by the above results, we present several characterizations of the strong metric subregularity of the Mordukhovich subdifferential for an extended-real-valued l.s.c. prox-regular and subdifferentially continuous function acting on an Asplund space.

Section 2 contains necessary definitions and facts. In Section 3 we extend the known equivalences for strong metric subregularity of subdifferentials for convex functions to the class of prox-regular and subdifferentially continuous functions defined on the Asplund spaces (Theorem 12). Theorem 15 is concerning the relationship between the strong metric subregularity and contingent derivative of the Mordukhovich subdifferential.

2. Preliminaries

Throughout this paper, and are Asplund spaces, that is, Banach spaces such that every separable subspace has a separable dual, and is the dual space of . The symbol always denotes the convergence relative to the norm while the symbol signifies the weak* convergence in the dual space . The closed ball centered at of radius is denoted by while the closed unit and dual unit balls are denoted by and , respectively. The distance function associated with a nonempty set is defined by
Let be a set-valued mapping between Banach spaces and . The domain and graph are defined by
respectively. For a mapping between a Banach space and its dual we define the sequential Painlevé-Kuratowski outer limit by

Definition 1. Given a subset and a point , the contingent cone of at is defined by

In stability analysis, the concept of metric regularity and its variants plays an important role; for more details and further references see, for example, [6–8]. Our study is focused on two key notions: metric subregularity and strong metric subregularity. They are defined as follows.

Definition 2. A mapping is said to be metrically subregular at for if and there is a positive constant along with neighborhoods of and of such that

Definition 3. A mapping is said to be strongly (metrically) subregular at for if and there is a positive constant along with neighborhoods of and of such that

We notice that the strong metric subregularity of at for is equivalent to the metric subregularity if is an isolated point of . The definition of metric subregularity can be simplified in the following way:
For a possibly smaller neighborhood of , see [1] for details. Likewise, the definition of strong subregularity can be simplified as

For an extended-real-valued function we define the domain of to be

In this paper, we assume that all the extended-real-valued functions are proper, that is, not identically equal to , and lower semicontinuous (l.s.c.) on .

The fundamental tools for studying general nonsmooth function are subdifferentials. The following two subdifferential notions are used in this paper.

Definition 4. Consider and . (i) is a proximal subgradient if there is with
for any from some neighborhood of . The proximal subdifferential of at is the collection of all proximal subgradients of at .(ii)The (basic, limiting, and Mordukhovich) subdifferential of at is
where the symbol means that with .

When is convex, is the usual subgradient set of convex analysis. When is smooth, is the singleton .

Definition 5. Let and . We say that is prox-regular at for if there exist and such that
whenever , , and with . If this holds for every , we say that is prox-regular at .

Definition 6. A function is subdifferentially continuous at for if for every there exists such that whenever and with some . If this occurs for all , we say that is subdifferentially continuous at .

The concepts of prox-regularity and subdifferential continuity were introduced in [9] and were further studied in [10, 11]. It was shown in those works that the class of functions with these properties is quite large. It includes functions, l.s.c. proper convex functions, and many functions that typically might be encountered in finite-dimensional optimization.

The well-known Ekeland’s variational principle [12] plays an important role in this paper.

Theorem 7 (Ekeland’s variational principle). Let be a complete metric space and a proper lower semicontinuous function bounded from below. Suppose that, for some and some ,
Then, for every there exists some point such that

In this section, we will give characterizations of the strong metric subregularity of the Mordukhovich subdifferential of extended-real-valued prox-regular subdifferentially continuous functions. Namely, we provide some equivalent conditions to the strong metric subregularity of subdifferentials of such functions including quadratic growth properties and locally strongly monotone of subdifferentials.

The following theorem is implied in [5, Corollary 3.3]; we present a slightly different proof.

Theorem 8. Let and . Suppose that the subdifferential is strongly metrically subregular at with modulus . Then, the following assertions are equivalent: (i)there are real numbers and such that
(ii)for any real number , there is a real number such that

Proof. Implication holds trivially.Next, we focus on . Since the subdifferential is strongly metrically subregular at with modulus , there is some positive constant such that
Let .We start by showing that
for all , where . Suppose that (19) is not true and there is some such that
This together with (16) implies that and
Hence, there is some slightly smaller than
such that
Since is l.s.c. on , by Ekeland’s variational principle (Theorem 7), there exists some such that
and for all
This implies that minimizes the function
over and
Then, Fermat’s stationary rule along with the subdifferential sum rule ([8, Theorem 3.6]) implies the inclusions
Therefore,
Additionally, since
one has
and thus
This strict inequality contradicts (18), since by (27).By induction we may then construct from (19) a strictly positive sequence satisfying
with for and . Consequently it gives
which completes the proof.

Corollary 9. Let and . Suppose that the subdifferential is strongly metrically subregular at with modulus . Then, the following assertions are equivalent: (i)there are real numbers and such that
(ii)for any real number , there is a real number such that

Proof. By [5, Corollary 3.3] and Theorem 8, we have that the equivalence holds.

The following theorem provides a characterization of strong metric subregularity of the subdifferential for a prox-regular and subdifferentially continuous function.

Theorem 10. Given and a pair , let be prox-regular at for with and subdifferentially continuous at for . Then, is strongly metrically subregular at with modulus satisfying if and only if for any real number , there is a real number such that

Proof. Suppose that is strongly metrically subregular at with modulus satisfying . Since is prox-regular at for with and subdifferentially continuous at for , by Definitions 5 and 6, there exists a real number such that
Hence, by Corollary 9, we have that inequality (37) holds.To justify the converse implication, assume that (37) holds for some and . Since is prox-regular at for with and subdifferentially continuous at for , there is a constant such that
for all . Fix any point ; if , it clearly follows that . Hence, we may suppose that ; it follows from (37) and (39) that
which implies that
By , the latter implies that , which justifies the strong metric subregularity of at and thus completes the proof of the theorem.

Definition 11 ([4, Definition 3.4]). Let . The point is locally strongly monotonically related to gph if there exist neighborhoods of and of together with a constant such that

Now we are going to characterize the strong metric subregularity of the Mordukhovich subdifferential for an extended-real-valued l.s.c. prox-regular and subdifferentially continuous function acting on an Asplund space. Let us note that the equivalences of statements (i)–(iv) have been already established in [4, Theorem 3.6] in the convex setting. We show the equivalences of statements (i)–(iv) in nonconvex case, and we prove that they are also equivalent to statement (v).

Theorem 12 (strong metric subregularity of the subdifferential). Let and . If is prox-regular at for with and subdifferentially continuous at for , then the following statements are equivalent:(i)the subdifferential is strongly metrically subregular at with modulus satisfying ,(ii)for any real number , there is an such that
(iii)there exist neighborhoods of and of and such that
(iv)the point is locally strongly monotonically related to gph ,(v)for any , there is a neighborhood of such that, for all and ,

Proof. The equivalence has been already proved in Theorem 10.To show the implication suppose that holds for some and . Since is prox-regular at for with and subdifferentially continuous at for , there is an such that
for all , .By letting equal , we obtain
It follows from (43) that
Thus,
and holds with .The implication is straightforward.To show the implication let us assume that holds. There are some neighborhoods of and of and a positive such that
Pick any ; if we are done. Otherwise, for any , we have
which implies that
that is, the subdifferential is strongly metrically subregular at .Finally, we prove the equivalence . Suppose that holds. Let , , and . From (45) we get
that is,
Moreover, by (46),
which together with inequality (54) gives
Letting in this inequality, we obtain (43).Suppose now that holds. Let and . Then, (56) holds. This together with (55) implies (45) which completes the proof.

Let us note that, by Fermat’s stationary rule, if a point is a local minimizer of the function , then we have . By Theorem 12, is strongly metrically subregular at for with modulus if and only if satisfies the following quadratic growth condition:
where is a positive number.

The following result on the strong subregularity of the subdifferential of the sum of two functions is a straightforward consequence of Theorem 12.

Corollary 13. Let , , and . Assume that is prox-regular at for with the constant and subdifferentially continuous at for . Assume that is prox-regular at for with the constant and subdifferentially continuous at for . If is strongly metrically subregular at with modulus , and is strongly metrically subregular at with modulus , , then is strongly metrically subregular at for .

Proof. By Theorem 12, the strong metrical subregularity of at with modulus yields, for any , the existence of a real number such that
The strong metrical subregularity of at with modulus implies that, for any , the existence of a real number such that
It follows from (58) and (59) that
where .Since is prox-regular at for with and subdifferentially continuous at for and is prox-regular at for with and subdifferentially continuous at for , there is a constant such that
for all .It follows from (61) and (62) that
for all .Inequality (60) together with (63) gives
for all .Applying Theorem 12 again, is strongly metrically subregular at for , which completes the proof.

Next we investigate relationships between the strong metric subregularity and contingent derivative which is a graphical concept of derivative for set-valued maps and was introduced by Aubin in [13].

Definition 14. Let and . The contingent derivative of at is a set-valued mapping defined as
Note that when is twice (Fréchet) differentiable (see, e.g., [14, Proposition ]). For more details on the contingent derivative, one can refer to the comprehensive monograph [14] by Aubin and Frankowska.

Theorem 15. Given and a pair , let be prox-regular at for with and subdifferentially continuous at for . Consider the following two statements: (i)the subdifferential is strongly metrically subregular at with modulus satisfying ,(ii)there is a constant such that is positive-definite with modulus in the sense that
Then, implication holds. Furthermore, the converse implication also holds if in addition dim .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Ewa Bednarczuk for her helpful discussions on improving the quality of the paper. The research of J. J. Wang was partly supported by the Foundation of Heilongjiang Provincial Educational Department (12521147). The research of W. Song was supported in part by the National Natural Sciences Grant (no. 11371116).